Quantum Information Processing

Global phase. Two quantum states are indistinguishable if they differ only by a global
phase. That is, |ψ〉 and e iφ |ψ〉 are in essence the same state. The global phase
difference is the factor e iφ . The equivalence of the two states is apparent from the fact
that their density matrices are the same.
Hilbert space. An n-dimensional Hilbert space consists of all complex n-dimensional
vectors. A defining operation in a Hilbert space is the inner product. If the vectors are
thought of as column vectors, then the inner product 〈x, y〉 of x and y is obtained by
forming the conjugate transpose x † of x and calculating 〈x, y〉 = x † y. The inner
product induces the usual squared norm |x| 2 = 〈x, x〉.
Information. Something that can be recorded, communicated, and computed with.
Information is fungible; that is, its meaning can be identified regardless of the
particulars of the physical realization. Thus, information in one realization (such as
ink on a sheet of paper) can be easily transferred to another (for example, spoken
words). Types of information include deterministic, probabilistic, and quantum
information. Each type is characterized by information units, which are abstract
systems whose states represent the simplest information of each type. The
information units define the “natural” representation of the information. For
deterministic information, the information unit is the bit, whose states are symbolized
by and . Information units can be put together to form larger systems and can be
processed with basic operations acting on few of them at a time.
Inner product. The defining operation of a Hilbert space. In a finite dimensional
Hilbert space with a chosen orthonormal basis {e i :1 ≤ i ≤ n}, the inner product of
two vectors x = Σ i x i e i and y = Σ i y i e i is given by Σ i x i y i . In the standard column
representation of the two vectors, this is the number obtained by computing the
product of the conjugate transpose of x with y. For real vectors, that product agrees
with the usual “dot” product. The inner product of x and y is often written in the form
〈x, y〉. Pure quantum states are unit vectors in a Hilbert space. If |φ〉 and |ψ〉 are two
quantum states expressed in the ket-bra notation, their inner product is given by
(|φ〉) † 〈ψ| = 〈φ|ψ〉.
Ket. A state expression of the form |ψ〉 representing a quantum state. Usually, |ψ〉 is
thought of as a superposition of members of a logical state basis |i〉. One way to think
about the notation is to consider the two symbols | and 〉 as delimiters denoting a
quantum system and ψ as a symbol representing a state in a standard Hilbert space.
The combination |ψ〉 is the state of the quantum system associated with ψ in the
standard Hilbert space via a fixed isomorphism. In other words, one can think of
ψ ↔|ψ〉 as an identification of the quantum system’s state space with the standard
Hilbert space.
Linear extension of an operator. The unique linear operator that implements a map
defined on a basis. Typically, we define an operator U on a quantum system only
on the logical states U : |i〉→|ψ i 〉. The linear extension is defined by U(Σ i α i |i〉) = Σ i α i |ψ i 〉.
Logical states. For quantum systems used in information processing, the logical states
are a fixed orthonormal basis of pure states. By convention, the logical basis for
qubits consists of |〉 and |〉. For larger dimensional quantum systems, the logical
basis is often indexed by integers, |0〉, |1〉, |2〉, and so on. The logical basis is often
called the computational basis, or sometimes, the classical basis.
Measurement. The process used to extract classical information from a quantum
system. A general projective measurement is defined by a set of projectors P i ,
satisfying Σ i P i = 1 and P i P j = δ ij P i . Given the quantum state |ψ〉, the outcome of a
measurement with the set {P i } i , is one of the classical indices i associated with a
projector P i . The index i is the measurement outcome. The probability of outcome i
is p i = |P i |ψ i 〉| 2 , and given outcome i, the quantum state “collapses” to P i |ψ i 〉/√p i .
Quantum Information Processing
Number 27 2002 Los Alamos Science 35